Differential Equations - A. B. Vasil'eva

. .

Résumé :
Translated from the Russian by Sossinskij, A.B.

Sommaire:
I. Introduction.- ? 1. The Concept of a Differential Equation.- ? 2. Physical Problems Leading to Differential Equations.- II. General Theory.- ? 1. Elementary Integration Methods.- ? 2. Theorems on the Existence and Uniqueness of the Solution of the Initial Value Problem for a First Order Equation Resolved with Respect to the Derivative. The Euler Polygonal Line Algorithm.- ? 3. Equations not Resolved with Respect to the Derivative.- ? 4. Existence and Uniqueness Theorems for the Solution of Normal Systems.- ? 5. Dependence of Solutions on Initial Values and Parameters.- ? 6. The Method of Successive Approximations (Picard's Method).- ? 7. The Contraction Mapping Theorem.- III. Linear Differential Equations.- ? 1. The Pendulum Equation as an Example of a Linear Equation. The Main Properties of Linear Equations with Constant Coefficients.- ? 2. General Properties of n-th Order Equations.- ? 3. Homogeneous n-th Order Linear Equations.- ? 4. Non-homogeneous Linear n-th Order Equations.-? 5. Linear n-th Order Equations with Constant Coefficients.- ? 6. Systems of Linear Equations. General Theory.- ? 7. Systems of Linear Differential Equations with Constant Coefficients.- ? 8. The Solutions in Power Series Form of Linear Equations.- IV. Boundary Value Problems.- ? 1. Formulation of Boundary Value Problems and their Physical Meaning.- ? 2. Non-homogeneous Boundary Value Problems.- ? 3. Eigenvalue Problems.- V. Stability Theory.- ? 1. Statement of the Problem.- ? 2. Study of Stability in the First Approximation.- ? 3. The Method of Lyapunov Functions.- ? 4. The Study of Trajectories in a Neighbourhood of a Stationary Point.- VI. Numerical Methods for the Solution of Ordinary Differential Equations.- ? 1. Numerical Methods for Solving Initial Value Problems.- ? 2. Boundary Value Problems.- VII. Asymptotics of Solutions of Differential Equations with Respect to a Small Parameter.- ? 1. Regular Perturbations.- ? 2. Singular Perturbations.- VIII. First Order Partial Differential Equations.- ? 1. Linear Equations.- ? 2. Quasilinear Equations.